Run modal operator while tool is selected
This is a hack--but for now (in 2.80/2.81) you can find out when a custom tool is activated and deactivated with a custom gizmo group:

class MYADDONNAME_TOOL_mytool(bpy.types.WorkSpaceTool):
    bl_idname = "myaddonname.mytool"
    bl_space_type='VIEW_3D'
    bl_context_mode='OBJECT'
    bl_label = "My tool"
    bl_icon = "ops.transform.vertex_random"
    bl_widget = "MYADDONNAME_GGT_mytool_activated"

class MYADDONNAME_GGT_mytool_activated(bpy.types.GizmoGroup):
    bl_label = "(internal)"
    bl_space_type = 'VIEW_3D'
    bl_region_type = 'WINDOW'
    bl_options = {'3D'}

    @classmethod
    def poll(cls, context):
        return True

    def setup(self, context):
        print("My tool activated!")

    def __del__(self):
        print("My tool deactivated!")

bpy.utils.register_class(MYADDONNAME_GGT_mytool_activated)
bpy.utils.register_tool(MYADDONNAME_TOOL_mytool, separator=True)


This isn't enough to invoke a modal operator as you're asking for, because the setup() method gets called in a drawing/rendering context, not one where you can just call bpy.ops.whatever('INVOKE_DEFAULT'). But this might be a useful starting point.